Beukers’ integrals and Apéry’s recurrences

Lalit Jain Faculty of Mathematics University of Waterloo Waterloo, Ontario N2L 3G1 CANADA

                      lkjain@uwaterloo.ca

Pavlos Tzermias Department of Mathematics University of Tennessee Knoxville, TN 37996-1300 USA

                     tzermias@math.utk.edu

Abstract We give a new and completely elementary proof of the fact that the rational approximations to π^{2} obtained by Apéry in his famous proof of the irrationality of certain values of the Riemann zeta function are identical to those obtained by Beukers in one of his alternative proofs of Apéry’s result.

1 Introduction

Apéry’s famous proof ([3], [10]) of the irrationality of ζ(3) makes ingenious use of certain identities and specific recurrences. The proof gives explicit rational approximations to ζ(3) which converge fast enough to prove its irrationality. Apéry’s proof also produces analogous rational approximations to ζ(2) = π^{2}/6, whose irrationality (in fact, transcendence) is wellknown. Specifically, in the case of ζ(2), Apéry considers the recurrence relation

(n + 1)^{2}u_{n+1} − (11n^{2} + 11n + 3)u_{n} − n^{2}u_{n−1} = 0.

Let {a_{n}} be the sequence solving the above recurrence with initial values a_{0} = 0 and a_{1} = 5. Also let {b_{n}} be the sequence solving the recurrence with initial values b_{0} = 1 and b_{1} = 3. Then the sequence {a_{n}/b_{n}} converges to ζ(2). Explicit formulas for a_{n} and b_{n} are given in [3] and [10]. We also note that {b_{n}} is the sequence A005258 in the On-Line Encyclopedia of Integer Sequences. Apéry also gives analogous arguments for ζ(3).

Shortly after Apéry announced his proof, Beukers ([6]) produced an elegant and entirely different proof of the irrationality of ζ(2) and ζ(3). In the case of ζ(2), Beukers considers the double integrals I_{n} defined by

I_{n} = ∫_{0}^{1} ∫_{0}^{1} \frac{(1 − y)^{n}P_{n}(x)}{1 − xy}dxdy,

where n ∈ ℕ and P_{n}(x) is the Legendre-type polynomial given by

P_{n}(x) = \frac{1}{n!} \frac{d^{n}}{dx^{n}} (x^{n}(1 − x)^{n}).

He then shows that

I_{n} = β_{n}ζ(2) − α_{n},

where α_{n}, β_{n} ∈ ℚ, for all n. An estimation of the latter linear form shows that it tends to 0 (as n approaches infinity) fast enough to yield the irrationality of ζ(2). Beukers also gives an analogous argument for the case of ζ(3) by using a triple integral instead. It is worth noting that an even more striking proof of Apéry’s result was given by Beukers in [4] using modular forms.

It is rather remarkable (and apparently known, as explained below) that the rational approximations to ζ(2) and ζ(3) obtained by Beukers are identical to those obtained by Apéry. We list below some places we were able to find in the literature where proofs of this fact (or of related facts) are given: For the 1-dimensional analogue of Beukers’ method (i.e., appropriate single-variable integrals), a related fact was established by Alladi and Robinson in [1], using properties of values of Legendre polynomials. The method is also discussed independently by Beukers in [5]. For the 2-dimensional and 3-dimensional analogues of Beukers’ method (i.e., Beukers’ double and triple integrals), the fact is verified by Nesterenko in [9], by using rather advanced arguments involving, among other things, contour integrals of Barnes type and transformation properties of hypergeometric series (see also the recent preprint by Zudilin ([12]). The reader may also consult the article by Fischler ([7]) for a complete survey of the subject.

The purpose of this paper is to give a new, short and completely elementary proof of the fact mentioned above for ζ(2).

Theorem 1.1 With notation as above, we have

a_{n} = α_{n}, b_{n} = β_{n},

for all n ∈ ℕ.

As the referee of an earlier version of this paper pointed out, another elementary proof of the same fact can be given by combining clever manipulations with Zeilberger’s powerful program Ekhad. Our proof is along different lines.

Our attempts to apply similar elementary arguments to the case of ζ(3) have invariably (and not surprisingly) led us to certain expressions involving special values of generalized hypergeometric series, which are notoriously difficult to compute. We do not address the case of ζ(3) any further in this paper.

The above theorem easily implies a recurrence relation between special values of certain generalized hypergeometric series (see the corollary below). It is not unlikely that this may also follow from the contiguous relations of Kummer (which were generalized by Wilson in [11]); we have not attempted to verify this. The reader may also consult the books by Andrews, Askey and Roy ([2]) or by Magnus, Oberhettinger and Soni ([8]) for a wealth of information regarding special functions of hypergeometric type.

If a is a positive integer, let _{3}F_{2}(a, a, a; 2a, 2a; 1) denote the value of the generalized hypergeometric series

3F_{2}(a, a, a; 2a, 2a; x) = 1 + ∑_{k=1}^{∞} \frac{(a … (a + k − 1))^{3}}{((2a) … (2a + k − 1))^{2}} \frac{x^{k}}{k!}

at x = 1. Then

Corollary 1.1 For every integer a such that a ≥ 2, we have

3F_{2}(a + 1, a + 1, a + 1; 2a + 2, 2a + 2; 1)

= −\frac{176a^{4} − 84a^{2} + 4a + 12}{a^{4}} _{3}F_{2}(a, a, a; 2a, 2a; 1)+

+\frac{256a^{4} − 128a^{2} + 16}{a^{4}} _{3}F_{2}(a − 1, a − 1, a − 1; 2a − 2, 2a − 2; 1).

2 The Proof

We first point out that some of the integrals below are improper; their use can be justified by replacing ∫_{0}^{1} by ∫_{ϵ}^{1−ϵ} and letting ϵ tend to 0. Also, in what follows, our manipulations of the series involved are valid because of their absolute and/or uniform convergence.

First note that

I_{0} = ∫_{0}^{1} ∫_{0}^{1} \frac{1}{1 − xy} dxdy = ∑_{k=0}^{∞} ∫_{0}^{1} ∫_{0}^{1} x^{k}y^{k}dxdy = ζ(2),

I_{1} = ∫_{0}^{1} ∫_{0}^{1} \frac{(1 − y)(1 − 2x)}{1 − xy}dxdy

= ∑_{k=0}^{∞} ∫_{0}^{1} ∫_{0}^{1}(x^{k}y^{k} − x^{k}y^{k+1} − 2x^{k+1}y^{k} + 2x^{k+1}y^{k+1})dxdy = −5 + 3ζ(2),

so the theorem is true for n = 0 and n = 1. As Beukers shows in [6], we have

I_{n} = (−1)^{n} ∫_{0}^{1} ∫_{0}^{1} \frac{x^{n}y^{n}(1 − x)^{n}(1 − y)^{n}}{(1 − xy)^{n+1}} dxdy,

for all n. Now, n-fold differentiation of the geometric series identity

\frac{1}{1 − u} = ∑_{k=0}^{∞} u^{k}

gives the following formal identity for n ∈ ℕ and an indeterminate u:

\frac{1}{(1 − u)^{n+1}} = ∑_{k=0}^{∞} \frac{(n + k)!}{k! n!}u^{k}.

Therefore,

I_{n} = (−1)^{n} ∑_{k=0}^{∞} \frac{(n + k)!}{k! n!} ∫_{0}^{1} ∫_{0}^{1} x^{n+k}y^{n+k}(1 − x)^{n}(1 − y)^{n}dxdy =

(−1)^{n} ∑_{k=0}^{∞} \frac{(n + k)!}{k! n!}(B(n + k + 1, n + 1))^{2},

where B(·, ·) denotes Euler’s beta function. Therefore,

I_{n} = (−1)^{n} ∑_{k=0}^{∞} \frac{(n + k)!^{3} n!}{(2n + k + 1)!^{2} k!}.

Since ζ(2) is irrational and {a_{n}}, {b_{n}} satisfy the same recurrence relation (with different initial conditions), it follows that, in order to prove the theorem, it suffices to show that the sequence {I_{n}} satisfies Apéry’s recurrence relation, i.e., we need to show that

(n + 1)^{2}I_{n+1} − (11n^{2} + 11n + 3)I_{n} − n^{2}I_{n−1} = 0,

for all n ≥ 1. Fix such an n. It suffices to show that

∑_{k=0}^{∞}((n + 1)^{2} \frac{(n + k + 1)!^{3} (n + 1)!}{(2n + k + 3)!^{2} k!} + (11n^{2} + 11n + 3)\frac{(n + k)!^{3} n!}{(2n + k + 1)!^{2} k!}

−n\frac{(n + k − 1)!^{3} n!}{(2n + k − 1)!^{2} k!}) = 0.

Let S_{k,n} denote the expression inside the infinite sum on the left-hand side of the above equality. A tedious calculation shows that

S_{k,n} = \frac{(n + k − 1)!^{3} n!}{(2n + k + 3)!^{2} k!}(−nk^{8} + (3 − n − 5n^{2})k^{7} + (10n^{3} + 66n^{2} + 88n + 31)k^{6}+

+(119n^{4} + 563n^{3} + 881n^{2} + 548n + 114)k^{5}+
+(314n^{5} + 1749n^{4} + 3479n^{3} + 3128n^{2} + 1268n + 183)k^{4}+
+(340n^{6} + 2412n^{5} + 6121n^{4} + 7310n^{3} + 4330n^{2} + 1179n + 109)k^{3}+
+(71n^{7} + 1119n^{6} + 4172n^{5} + 6737n^{4} + 5364n^{3} + 2043n^{2} + 291n)k^{2}+
+(−138n^{8} − 503n^{7} − 349n^{6} + 782n^{5} + 1468n^{4} + 885n^{3} + 183n^{2})k
−(79n^{9} + 474n^{8} + 1141n^{7} + 1400n^{6} + 913n^{5} + 294n^{4} + 35n^{3})).

Let T_{m,n} denote the m-th partial sum of the latter infinite series, i.e.,

T_{m,n} = ∑_{k=0}^{m} S_{k,n}.

We claim that T_{m,n} is given by the following closed formula:

T_{m,n} = \frac{(n + m)!^{3} n!}{(2n + m + 3)!^{2} m!}(m^{6} + (4n + 9)m^{5} − (13n^{2} − 7n − 26)m^{4}

−(102n^{3} + 228n^{2} + 112n − 15)m^{3} − (225n^{4} + 822n^{3} + 1025n^{2} + 479n + 52)m^{2}
−(217n^{5} + 1057n^{4} + 1957n^{3} + 1691n^{2} + 658n + 84)m
−(79n^{6} + 474n^{5} + 1141n^{4} + 1400n^{3} + 913n^{2} + 294n + 35)).

Although this formula is difficult to guess, its proof is a tedious but straightforward induction argument (on m), using the explicit formula for S_{k,n} given above. The authors suspected the existence of such a closed formula for T_{m,n} after explicitly computing it for the first few values of m. It should be pointed out that there is an algorithm (due to Gosper, and lying at the heart of the Ekhad program) that, given a hypergeometric summand S_{k}, determines whether or not ∑_{k=0}^{m} S_{k} has a hypergeometric closed form. Also, one may try to use the Wilf-Zeilberger algorithm of creative telescoping to compute T_{m,n}; we have not attempted to use any of these algorithms.

It now remains to show that T_{m,n} tends to 0 as m approaches infinity. By Stirling’s formula, we see that

lim_{m→∞} T_{m,n} = lim_{m→∞} \frac{(\frac{n+m}{e})^{3n+3m}}{(\frac{m}{e})^{m}(\frac{2n+m+3}{e})^{4n+2m+6}} \frac{\sqrt{8π^{3}(n + m)^{3}}}{2π(2n + m + 3)\sqrt{2πm}}m^{6}

= lim_{m→∞} m^{−n}e^{n+6} = 0,

and this completes the proof of the theorem.

Now note that the formula for I_{n} given in our proof of the theorem may be restated as follows:

I_{n} = (−1)^{n} \frac{n!^{4}}{(2n + 1)!^{2}} _{3}F_{2}(n + 1, n + 1, n + 1; 2n + 2, 2n + 2; 1).

Since I_{n} satisfies Apéry’s recurrence, the corollary follows from an easy calculation.

3 Acknowledgments

This work was done during the first author’s participation in the Research Experiences for Undergraduates program at the University of Tennessee in the Summer of 2004. We thank Suzanne Lenhart for organizing the program and the National Science Foundation for providing financial support for it. We also thank David Dobbs and the referee for their suggestions on a previous version of the manuscript.

References

[1] K. Alladi and M. Robinson, On certain irrational values of the logarithm, Number Theory, Carbondale 1979, Lecture Notes in Math. 751, Spinger, Berlin 1979, 1–9.

[2] G. Andrews, R. Askey and R. Roy, Special Functions, Encyclopedia of Mathematics and its Applications, 71, Cambridge University Press, 1999.

[3] R. Apéry, Irrationalité de ζ(2) et ζ(3), Astérisque 61 (1979), 11–13.

[4] F. Beukers, Irrationality proofs using modular forms, Astérisque 147-148 (1987), 271-283.

[5] F. Beukers, Legendre polynomials in irrationality proofs, Bull. Austral. Math. Soc. 22 (1980), 431–438.

[6] F. Beukers, A note on the irrationality of ζ(2) and ζ(3), Bull. London Math. Soc. 11 (1979), 268–272.

[7] S. Fischler, Irrationalité de valeurs de zêta (d’après Apéry, Rivoal, …), Séminaire Bourbaki 2002-2003, exposé no. 910, to appear in Astérisque.

[8] W. Magnus, F. Oberhettinger and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, Third enlarged edition, Grund. der math. Wissen. 52, Springer-Verlag, New York, 1966.

[9] Y. Nesterenko, Integral identities and constructions of approximations to zeta-values, J. Théor. Nombres Bordeaux, 15 (2003), 535-550.

[10] A. van der Poorten, A proof that Euler missed... Apéry’s proof of the irrationality of ζ(3), Math. Intelligencer 1 (1978/79), 195–203.

[11] J. Wilson, Hypergeometric Series, Recurrence relations and Some New Orthogonal polynomials, Ph.D. Thesis, University of Wisconsin, Madison, 1978.

[12] W. Zudilin, Approximations to -, di- and tri- logarithms, preprint, posted at

    http://front.math.ucdavis.edu/math.CA/0409023, September 2, 2004.

2000 Mathematics Subject Classification: Primary 11J72; Secondary 11B83, 11Y55, 33C20 . Keywords: Beukers’ integrals, Apéry’s recurrences.

(Concerned with sequence \underline{A005258}.)

Received September 6 2004; revised version received January 3 2005. Published in Journal of Integer Sequences, January 10 2005.

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